Which Telekom router is it? The smart3 also supports mesh. Get it if it is rented. Then connect the speedhome wifi as an access point directly to the LAN cable and everything will be fine!
Those that you connect directly to the network cable are called Access Points, not Repeaters.
Do you have additional network ports where you can connect Access Points? Otherwise, just connect a small switch where the TV is connected, and then connect the TV + Access Point there. Ideally, this works on every floor.
A good source is Ubiquiti. Their Unifi Access Points are great.
Tell me, would this be the right one?
Ubiquiti Networks 2.4GHz/5GHz 867mbits?
Is it then like a new router, do devices automatically switch between the Telekom router and this one?
How and where the access points are is relatively irrelevant (without wanting to start a fundamental discussion here). If the concrete ceilings etc. interfere, then one access point per floor...on the wall or ceiling or on the shelf...done.
Sounds really good, do the access points have disadvantages?
Tell me, would this be the right one?
Ubiquiti Networks 2.4GHz/5GHz 867mbits?
Is it then like a new router, do the devices automatically switch between the Telekom router and this one?
The description is not specific enough. I would recommend the UniFi Access Point AC Nano HD.
It works best with identical devices with identical settings and Wi-Fi turned off on the Telekom router. With mixed devices, I am not sure how well the switching works.
If you have not had network outlets installed for access points, then I think the existing LAN outlets are near the floor. The antennas of the Unifi access points are designed for ceiling mounting. You can also nail them to the wall, but that is not quite optimal. Since you have a Telekom router, I would use a Telekom Speed Home Wi-Fi with it and operate it together with the router.